On additive functions with values in Abelian groups

Abstract. Let \({\mathbb G}_0\subseteq {\mathbb G}\) be Abelian groups. We prove that if \(\Gamma\in{\mathbb G}\), \(f_0,f_1,f_2,f_3,f_4,f_5\) are \({\mathbb G}\)-valued completely additive functions and \[ \sum_{j=0}^{5}f_j(n+j)+\Gamma\in {\mathbb G}_0\quad \text{for all}\quad n\in {\Bbb Z}, \] then \(\Gamma\in{\mathbb G}_0\) and \(f_j(n)\in {\mathbb G}_0\) for all \(n\in {\Bbb Z},~j\in\{0,1,\cdots,5\}\).